A necessary and sufficient condition for a unique maximum with an application to potential games

• Published in: Economics letters Vol. 161; pp. 120 - 123
• Main Author: Christensen, Finn
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.12.2017; Elsevier Science Ltd
• Subjects: Boundary conditions; economics; Game theory; Games; Index theory; Optimization; Potential games; Regularity; Studies; Uniqueness; Potential games; C02; Optimization; C72; Index theory
• ISSN: 0165-1765; 1873-7374
• DOI: 10.1016/j.econlet.2017.10.008

Under regularity and boundary conditions which ensure an interior maximum, I show that there is a unique critical point which is a global maximum if and only if the Hessian determinant of the negated objective function is strictly positive at any critical point. Within the large class of Morse functions, and subject to boundary conditions, this local and ordinal condition generalizes strict concavity, and is satisfied by nearly all strictly quasiconcave functions. The result also provides a new uniqueness theorem for potential games. •In some contexts, “∇f(x∗)=0⇒det(−D2f(x∗))>0” iff there is a unique critical point that is a global maximum.•This is an alternative to strict quasiconcavity which is only a sufficient condition.•The result is applied to potential games and yields a new uniqueness theorem.•The proof is an application of the Poincaré–Hopf Theorem from differential topology